# On the $\text{PGL}_{2}$-invariant quadruples of torsion points of elliptic curves

## TL;DR

This paper classifies quadruples of torsion points on elliptic curves whose images under a standard double cover have a cross ratio independent of the curve, revealing special invariant configurations.

## Contribution

It provides a complete classification of PGL₂-invariant quadruples of torsion points on elliptic curves, a problem previously not fully resolved.

## Key findings

- Identifies all quadruples with invariant cross ratios across elliptic curves
- Establishes conditions for PGL₂-invariance of torsion point configurations
- Completes the classification of such invariant quadruples

## Abstract

Let $E$ be an elliptic curve and $\pi:E\to\mathbb{P}^{1}$ a standard double cover identifying $\pm P\in E$. It is known that for some torsion points $P_{i}\in E$, $1\leq i\leq4$, the cross ratio of $\{\pi(P_{i})\}_{i=1}^{4}$ is independent of $E$. In this article, we will give a complete classification of such quadruples.